The Divisibility Properties of Primary Lucas Recurrences with Respect to Primes

(4) vn = vl + r». To c o n t i n u e , we need t h e fo l lowing d e f i n i t i o n s which a r e modeled a f t e r t h e n o t a t i o n of Hal ton [3] . The l e t t e r p w i l l always denote a r a t i o n a l p r ime . Vd^nAJtton I : v ( a , b9 p) i s t h e numeric of t h e PR u(a9 b) modulo p . I t i s the number of n o n r e p e a t i n g terms modulo p . Vt^AJbitiovi 2°> ] i (a , b9 p) i s t he pe r i od of t h e PR u(a9 b) modulo p . I t i s t h e l e a s t p o s i t i v e i n t e g e r k such t h a t n+k = n (mod p) is true for all n >_ v(a, b9 p) . Clearly, if v(a9 bs p) = 0, M(aibfP) ° a n d n(a,&,P)+i 1 ( m o d p ) .